12009, Oct. 12thsupported by Global Center of Excellence Program“Nanoscience and Quantum Physics”

Origin of Resonances in Chiral Dynamics

Tokyo Institute of TechnologyaTetsuo Hyodoa

2

Contents

KN scattering and the Λ(1405) resonance

Introduction to chiral dynamicsStructure/origin of the Λ(1405) resonance

Summary

Contents

・Dynamical or CDD pole (quark state) ?・Nc Behavior and quark structure・Electromagnetic properties

・Isospin interference and πΣ spectrum・Double-pole Λ(1405) and πΣ spectrum

3

Physics of the Λ(1405)

mass : 1406.5 ± 4.0 MeV, width : 50 ± 2 MeVdecay mode: 100%

(PDG)

“naive” quark model : p-wave ~1600 MeV?

N. Isgur, G. Karl, PRD18, 4187 (1978)

Coupled channel multi-scattering

R.H. Dalitz, T.C. Wong,G. Rajasekaran, PR153, 1617 (1967)

MB

KN scattering and the Λ(1405) resonance

KN interaction below thresholdT. Hyodo, W. Weise, PRC 77, 035204 (2008)

--> KN potential, kaonic nucleiKN

KN scatt.?

energyπΣ

Λ(1405)A. Dote, T. Hyodo, W. Weise, NPA804, 197 (2008); PRC 79, 014003 (2009)

4

“Mass” of the Λ(1405)

PDG

R.H. Dalitz, and A. Deloff, J. Phys G17, 289 (1991)Analysis of the Hemingwayʼs data by phenomenological model with I=0 to extract mass and width.

Spectrum is not purely in I=0, but with some contamination.Analysis is valid only when |TΛ∗ |! |T I=0,1non-resonant|

(they knew the isospin interference and discussed it)

KN scattering and the Λ(1405) resonance

5

Isospin interference in πΣ spectrumTo select I=0 component, it is needed to observe all three πΣ charged states (π0Σ0, π±Σ∓) simultaneously.

KN scattering and the Λ(1405) resonance

The isospin interference is observed experimentally!The interference is strong enough to change the spectrum (peak position, width, size of cross section, etc.)!

CLAS, K. Moriya@HYP-X (2009)

γp→ K+Λ(1405)

6

Double-pole structure in chiral dynamicsPole of the scattering amplitude : resonance

KN scattering and the Λ(1405) resonance

Tij(√

s) ∼ gigj√s−MR + iΓR/2

∼

1360

1380

1400

1420

1440

-20

-40

-60

-80

0.5

1.0

1.5

0.5

1.0

1.5

!(1405)

Re[z]Im[z]

|T|

T. Hyodo, W. Weise, Phys. Rev. C 77, 035204 (2008)

Origin of the two poles change πΣ spectraD. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Nucl. Phys. A 723, 205 (2003)

7

Interference and change of the πΣ spectrumKN scattering and the Λ(1405) resonance

Schematic decomposition of πΣ amplitudeTπΣ ∼ T (Λ∗) + Tnon-resonant(I = 0) + Tnon-resonant(I = 1)

Spectral change in π+Σ-, π-Σ+, π0Σ0

8

Chiral symmetry breaking in hadron physics

Consequence of chiral symmetry breaking in hadron physics

Chiral symmetry and its breaking

Underlying QCD observed hadron phenomenaSU(3)R ⊗ SU(3)L → SU(3)V

- constraints on the interaction of NG boson-hadron low energy theorems

9

s-wave low energy interactionLow energy NG boson (Ad)- target hadron (T) scattering

Projection onto s-wave : Weinberg-Tomozawa termY. Tomozawa, Nuovo Cim. 46A, 707 (1966); S. Weinberg, Phys. Rev. Lett. 17, 616 (1966)

Cα,T = 〈2FT · FAd〉 = C2(T ) − C2(α) + 3

Group theoretical structure of the target and flavor SU(3) symmetry determines the sign and strength of the interaction

Cij =∑

α

Cα,T

(8 T α

IMi , YMi ITi , YTi I, Y

)(8 T α

IMj , YMj ITj , YTj I, Y

)

Low energy theorem: leading order term in ChPT

Introduction to chiral dynamics

Vij = −Cij4f2

(ωi + ωj) energy of π (derivative coupling)

decay constant of π (gV=1)

10

Scattering amplitude and unitarityUnitarity of S-matrix : Optical theorem

Im [T−1(s)] =ρ(s)2

phase space of two-body state

General amplitude by dispersion relation

T−1(√

s) =∑

i

Ri√s−Wi

+ ã(s0) +s− s0

2π

∫ ∞

s+ds′

ρ(s′)(s′ − s)(s′ − s0)

Scattering amplitude

V? chiral expansion of T, (conceptual) matching with ChPTT (1) = V (1), T (2) = V (2), T (3) = V (3) − V (1)GV (1), ...

Amplitude T : consistent with chiral symmetry + unitarity

Ri, Wi, a : to be determined by chiral interaction

Identify dispersion integral = loop function G, the rest = V-1

T (√

s) =1

V −1(√

s)−G(√

s; a)

Introduction to chiral dynamics

11

Description of hadron-NG boson scatteringOverview of chiral dynamics

Y. Tomozawa, Nuovo Cim. 46A, 707 (1966); S. Weinberg, Phys. Rev. Lett. 17, 616 (1966)

- Interaction

12

T. Hyodo, S.I. Nam, D. Jido, A. Hosaka, PRC68, 018201 (2003); PTP 112, 73 (2004)

200

150

100

50

0

!T [

mb]

300200100

K-p

70

60

50

40

30

20

10

0300200100

"0#

200

150

100

50

0300200100

"+$%

60

50

40

30

20

10

0

!T [

mb]

300200100Plab [MeV/c]

K0n

60

50

40

30

20

10

0300200100

Plab [MeV/c]

"0$0

80

60

40

20

0300200100

Plab [MeV/c]

"%$+

KN scattering : comparison with data

14401420140013801360s [MeV]

!" m

ass d

istrib

utio

n

γ Rc Rn

exp. 2.36 0.664 0.189

theo. 1.80 0.624 0.225

Total cross section of K-p scattering Branching ratio

πΣ spectrum

Good agreement with data above, at, and below KN thresholdΛ(1405) mass, width, couplings : prediction of the model

Λ(1405)

Introduction to chiral dynamics

13

Dynamical state and CDD poleResonances in two-body scattering- Knowledge of interaction (potential)- Experimental data (cross section, phase shift,...)

(b) CDD pole: elementary, independent, ...

(a) dynamical state: molecule, quasi-bound, ...

L. Castillejo, R.H. Dalitz, F.J. Dyson, Phys. Rev. 101, 453 (1956)

Resonances in chiral dynamics -> (a) dynamical?

Structure/origin of the Λ(1405) resonance

MB

... in the present case : meson-baryon molecule

... in the present case : three-quark state

14

CDD pole contribution in chiral unitary approachAmplitude in chiral unitary model

V : interaction kernelG : loop integral

We point out that the loop function G can contain the CDD pole contribution.

Λ(1405) in KN scattering --> mostly dynamical

T. Hyodo, D. Jido, A. Hosaka, Phys. Rev. C78, 025203 (2008).

N(1535) in πN scattering --> dynamical + CDD pole

!"#

!$#

!%#

!

#'()*)+,

-./

0$1#0$##011#01##0%1#0%##021#3-)*)+,-./

*0!4)5)!60%#17

*84)5)8601217

*&!4)5)!60%#17

T =1

V −1 −G

We propose “natural renormalization scheme” to exclude CDD pole contribution in G (subtraction constant).

Known CDD pole contribution : those in V

Structure/origin of the Λ(1405) resonance

15

Nc scaling in the modelNc : number of color in QCDHadron effective theory / quark structureThe Nc behavior is known from the general argument. non-qqq structure of the Λ(1405)

Structure/origin of the Λ(1405) resonance

16

Attaching photon to resonance--> em properties : rms, form factors,...

Electromagnetic properties

T. Sekihara, T. Hyodo, D. Jido, in preparation

--> form factor F(Q2), density distribution ρ(r)Computation at finite Q2 is now underway

Λ(1405) has spatially large size. c.f. neutron: -0.12 [fm2]

T. Sekihara, T. Hyodo, D. Jido, Phys. Lett. B669, 133-138 (2008)

Evaluated mean squared radii :|〈r2〉E| = 0.33 [fm2]

--> support the meson-baryon picture

Structure/origin of the Λ(1405) resonance

17

Summary : Λ(1405) and chiral dynamics

Spectrum of the Λ(1405) is affected by isospin interference. Precise data from J-lab/J-PARC will be crucial for the hadron/nuclear physics.

Chiral dynamics: chiral interaction + coupled-channel unitarity

Internal structure of resonances can be investigated in several ways.

We discuss the physics of the Λ(1405) and its description by chiral dynamics.

=> successful description of KN scattering + the Λ(1405) resonance

Summary

18

Dynamical or CDD?

Analysis of Nc scaling

Electromagnetic properties

The structure of the Λ(1405) is studied:Summary : Structure of Λ(1405)

=> dominance of the MB components

=> non-qqq structure

=> large e.m. sizeT. Sekihara, T. Hyodo, D. Jido

T. Hyodo, D. Jido, A. Hosaka

T. Hyodo, D. Jido, L. Roca

Summary

19

Dynamical or CDD?

Analysis of Nc scaling

Electromagnetic properties

Independent analyses consistently support the meson-baryon molecule picture for the Λ(1405)

The structure of the Λ(1405) is studied:

MB

Summary : Structure of Λ(1405)

=> dominance of the MB components

=> non-qqq structure

=> large e.m. sizeT. Sekihara, T. Hyodo, D. Jido

T. Hyodo, D. Jido, A. Hosaka

T. Hyodo, D. Jido, L. Roca

Summary